| import Mathlib |
| import OAI.Geometry.Cannon.PlaneTopology.HomeoPlane |
| import OAI.Geometry.Cannon.Boundary.BoundaryUniformComplementGraph |
| import OAI.Geometry.Cannon.Boundary.BoundaryUniformConnectedComplement |
| import OAI.Geometry.Cannon.PlaneTopology.ChartTameFaceCount |
| import OAI.Geometry.Cannon.PlaneTopology.BoundaryBallCover |
| import OAI.Geometry.Cannon.PlaneTopology.InteriorRadiusControl |
| import OAI.Geometry.Cannon.Boundary.BoundaryUniformRingPacking |
| import OAI.Geometry.Cannon.Uniformization.BoundaryUniformizationInputs |
| import OAI.Geometry.Cannon.Conformal.QuasiMobius |
| import OAI.Geometry.Cannon.Conformal.PlaneWeakQSBounded |
| import OAI.Geometry.Cannon.Actions.BoundaryAction |
|
|
| namespace OAI |
|
|
| /-! A group with a hyperbolic Cayley graph and spherical boundary admits a proper, cocompact action on hyperbolic three-space with finite kernel. -/ |
|
|
| section |
| noncomputable section |
| namespace CannonRelease |
| universe u |
|
|
| theorem cannon (G : Type u) [Group G] [TopologicalSpace G] [DiscreteTopology G] |
| (D : CayleyData G) (hthin : D.ThinTriangles) |
| (hsphere : Nonempty (Boundary D ≃ₜ SphereTwo)) : |
| ∃ ρ : G →* H3Isom, ProperAction ρ ∧ CocompactAction ρ ∧ |
| (ρ.ker : Set G).Finite := by |
| obtain ⟨δ,hδ⟩ := hthin |
| obtain ⟨φ⟩ := hsphere |
| let _ := visualMetricSpace D hδ |
| obtain ⟨ψ,hψ⟩ := boundary_quasiMobius_uniformizer D hδ φ |
| exact boundary_uniformized_action D hδ ψ hψ |
| end CannonRelease |
|
|
| end |
| end |
|
|
| end OAI |
|
|
